Exponential functions

An exponential function is one that has an unknown in place of the exponent.

Definition

An exponential function f has the form

$$f(x)=a^x,\quad a\in\mathbb{R},\quad a>0,\quad a\ne1$$

The symbol a is an arbitrary number, it is not a complex expression; it is called the base. The expression x is called the exponent. So the base can be, for example, an integer 3, a rational number $\frac12$, or a constant π.

Why do we put conditions on the base a? If a were equal to one, we would get a constant function because "one for anything" is always one. It is true that 1² = 1, 1⁹ = 1, 1⁶⁶⁶ = 1. And since we don't classify a constant function as an exponential, a must be different from one, so a≠1. By the same token, a≠0 must be true.

And why must a>0 hold ? Because the power itself is defined only for positive numbers. For example, if we choose one half for x, then if we calculate $a^{\frac12}$, it is the same as calculating the square root of a. Yet, we can only square root a non-negative number.

If we satisfy the previous conditions, the exponential function f(x) = a^x has a defining domain of real numbers.

Quadratic vs. exponential function

You can easily confuse the exponential function with the quadratic function. It is true that a quadratic function has an unknown as the base and a number as the exponent, while an exponential function has a number as the base and an unknown in the exponent. So, for example, quadratic functions are the following functions: x², x² + 4 or x² − x + 1. Exponential functions are the functions: 2^x, 4^x or π^x.

Important exponential functions

An important exponential function is the "natural exponential function". It is a function that has a base constant e, i.e., the Euler number. Euler number is an irrational number, i.e. a number with infinite decimal expansion, it cannot be exactly quantified. Its approximate value is e = 2,718 281 828… This function is often denoted as exp.

Another important exponential function is the "decadic exponential function", which has a base of ten: a = 10.

The form of the function as a function of the value of a

The function has two basic forms, differing depending on the value of the base. There are two intervals: (0,1) and (1, ∞). If a is from the first interval, i.e. is less than one, then the function is decreasing. If it is from the second interval, then it is increasing. This makes sense, of course. As an example, consider $a=\frac12$, i.e. the function $f(x)=\frac12^x$. What happens if we successively add a two, a three and a four after x? The value of the function will gradually decrease, because we get: $f(2)=\frac12^2 = \frac12 \cdot \frac12=\frac14$, then $f(x)=\frac12^3= \frac12 \cdot \frac12 \cdot \frac12$, which is equal to $\frac18$. For f(4), we get half of one-eighth, i.e. $\frac{1}{16}$. We get smaller and smaller numbers, because the base is less than one. If we multiply numbers that are less than one, we always get a smaller number.

Conversely, if a>1, then the result of the multiplication will be a larger number. If we have an exponential function f(x) = 3^x, we get four results for values of two, three: f(2) = 3² = 3 · 3 = 9. For three: f(3) = 3³ = 3 · 3 · 3 = 27. Then f(4) = 81. So the function is increasing. Note that even if the base is any other number that is greater than one, the function will still be increasing.

Therefore, we must also distinguish between two types of graphs depending on the value of the base a.

Graphs of exponential functions

We know from the previous chapter that a graph must have two basic forms depending on the value of the base a. First, we show the graph for the values of a of the interval (0, 1).

We can see that the function is decreasing and passes through the point [0, 1]. This is no coincidence - we know from the properties of powers that anything to zero is one. Thus $\frac12^0=1, 5^0=1, 33^0=1$ etc. Therefore, every curve of an exponential equation must also pass through this point. Even the curve that will fit the exponential function at a>1, as seen in the following figure:

Properties of exponential functions

• Thedefining domain is the set of all [real numbers](real numbers)
• The range of values is an interval (0, ∞)
• It is bounded from below, a^x>0. Unbounded from above.
• It has neither a maximum nor a minimum.

The monotonicity of the function then depends on the value of a. If a is from the interval (0, 1), then the function is decreasing. If it is from the interval (1, ∞), then it is increasing.

Exponential growth

The term "Exponential growth" is used quite often in common parlance. Usually what one means by it is that something is rising terribly fast. A classic case in point is the division of bacteria, as we know from the TV series Once Upon a Time. Once in a while there was a bad bacterium that started multiplying so that it always split. We can describe this method of multiplication by an exponential function f(x) = 2^x. The function value will give us the number of bacteria after x rounds of division.

Example: at the beginning (zero number of divisions, x = 0) we have one bacterium, i.e. 2⁰ = 1. After the first round of multiplication we have 2¹ = 2, i.e. two bacteria. This fits, one bacterium will divide into two. Now both of these bacteria will split, i.e. we have 2² = 4 four bacteria. Again, each one divides, so after the third round we have 2³ = 8 bacteria. And so on.

This multiplication, although it doesn't seem like it, is terribly fast. How many bacteria do we have after the tenth round? 2¹⁰ = 1024 That's a nice number. After 20 rounds, we have: 2²⁰ = 1 048 576 After thirty rounds, we're up to over a billion.

This rate is much higher than the ordinary quadratic equation. If we compare the speed of the exponential function f(x) = 2^x and the quadratic function g(x) = x², the exponential function clearly wins. We already know the value of the function f at the point x = 20, while the value of the quadratic function g at the point x = 20 is equal: g(20) = 20² = 400. This is many times less than 1 048 576.